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We study the number of linear extensions of a partial order with a givenproportion of comparable pairs of elements, and estimate the maximum andminimum possible numbers. We also consider a random interval partial order on$n$ elements, which has close to a third of the pairs comparable with highprobability: we show that the number of linear extensions is $n! \,2^{-\Theta(n)}$ with high probability.

Linear Extensions and Comparable Pairs in Partial Orders